Geometry and Representation Theory of Symplectic Resolutions

代数几何是研究多项式方程组的解集的学科。辛变异体是具有长度概念的变异体，因此在每个点周围变异体看起来像一个偶维向量空间。辛结构在经典力学中自然出现，经典力学是描述宏观物体运动的物理学分支。本研究项目的主要目标是利用线性代数和几何的工具来更好地理解和利用辛变量所享有的自然对称性，即辛分辨率。这项工作有望加强该领域与几何表示理论、组合学和弦理论的联系。更详细地说，这个研究项目由四个相互关联的子项目组成。第一个子课题是研究辛分辨对之间的辛对偶性。这种对偶性与三维规范理论中的镜像对偶性密切相关。第二个子课题是证明一个关于辛锥的交上同调与其辛分辨率的量子上同调的猜想。第三个子项目是用分区平铺法对高曲品种进行分类，就像用扇形法对环曲品种进行分类一样。第四个子项目是研究一个新的矩阵不变量，称为它的Kazhdan-Lusztig多项式。

Algebraic geometry is the study of solution sets of systems of polynomial equations called varieties. Symplectic varieties are varieties equipped with a notion of length so that around each point the variety looks like an even-dimensional vector space. Symplectic structures arise naturally in classical mechanics, the branch of physics that describes the motion of macroscopic objects. The main goals of this research project are to use tools from linear algebra and geometry to better understand and exploit natural symmetries enjoyed by symplectic varieties known as symplectic resolutions. The work is expected to strengthen connections of the field with geometric representation theory, combinatorics, and string theory.In more detail, this research project consists of four interrelated subprojects. The first subproject is to study symplectic duality between pairs of symplectic resolutions. This duality is closely related to mirror duality for 3-dimensional gauge theories. The second subproject is to prove a conjecture relating the intersection cohomology of a symplectic cone to the quantum cohomology of its symplectic resolution. The third subproject is to classify hypertoric varieties using zonotopal tilings, much in the same way that toric varieties are classified by fans. The fourth subproject is to study a new invariant of a matroid called its Kazhdan-Lusztig polynomial.